# Interpretation of coefficient on interactive effects term in regression analysis

I have two continuous variables: $$X1$$ and $$X2$$, both have a positive correlation on the dependent variable $$y$$ (continuous). I found that the interaction term $$(X1*X2)$$ is statistically significant for $$p\lt(0.00)$$ when added to the model. The final model is the following:

$$y=aX_1+bX_2-c(X_1*X_2)$$

Where $$a$$, $$b$$, and $$c$$ are the regression coefficients. However, I am wondering why the regression coefficient for $$c$$ of the interaction term is negative? What does it mean?

If I only add the interaction term $$(X1*X2)$$ to the model why is the coefficient positive when if I add it with $$(X1)$$ and $$(X2)$$ the regression coefficient becomes negative? It doesn't make sense!

If the following statement is true, Then it means y should increase as either X1 or X2 increases.

I have 2 continuous variables; X1 and X2, both have a positive correlation on the dependent variable y (continuous)

$$y=aX_1+bX_2-c(X_1*X_2)$$

but I wonder why the regression coefficient c of the interaction term is negative? what does it mean?

you have a negative sign before the c coefficient which means it tries to decrease y as X1,X2 increases but the data necessitates that it should increase. Hence c is negative.

why if I add only the interaction term (X1*X2) to the model, the coefficient is positive while if I add it with (X1) and (X2) the regression coefficient becomes negative? it doesn't make sense!

By adding just X1*X2 to your does it look like the equation below?

$$y=c(X_1*X_2)$$

Then it makes sense only if c is positive if not then Y will decrease with increase in X1, X2. It makes sense.

• Thank for your answer , is it mean that X1 is function of X2 and X2 is function of X1 ? Nov 12, 2020 at 22:48
• @YazanAlatoom No X1 and X2 are independent variables and Y is an independent variable. If you find the answer useful, then you could upvote or accept it Nov 13, 2020 at 10:41
• the final model is the following: Y=1.956+0.102*X1+0.830*X2-0.063*X1*X2 , the range of X1:1-10 , X2:0.01-6 , in some cases value of Y is lower when some value of X1 and X2 is high than when the value of X1 AND X2 is low Nov 13, 2020 at 10:52